Thanks Michael.

I also need an intercept-only negative binomial model with unknown scale 
parameter. So my thought was on borrowing some codes that already existed. I 
think Ivan's solution is an excellent one and can be extended to other 



On May 11, 2019, at 9:48 AM, Michael Weylandt 
<<>> wrote:

On Sat, May 11, 2019 at 8:28 AM Wang, Zhu 
<<>> wrote:

I am open to whatever suggestions but I am not aware a simple closed-form 
solution for my original question.

It would help if you could clarify your original question a bit more, but for 
at least the main three GLMs, there are closed form solutions, based on means 
of y. Assuming canonical links,

- Gaussian: intercept = mean(y)
- Logistic: intercept = logit(mean(y))  [Note that you have problems here if 
your data is all 0 or all 1]
- Poisson: intercept = log(mean(y)) [You have problems here if your data is all 

(Check my math on these, but I'm pretty sure this is right.)

Like I said above, this gets trickier if you add observation weights or 
offsets, but the same ideas work.

Stepping back to the statistical theory: GLMs predict the mean of y, 
conditional on x. If x doesn't vary (intercept only model), then the GLM is 
just predicting the mean of y and the MLE for the mean of y is exactly that 
under standard GLM assumptions - the sample mean of y.

We then just have to use the link function and its inverse to transform to and 
from the observation space (where mean(y) lives) and the linear predictor space 
(where the intercept term naturally lives).


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